[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 9 from proof intshiftright-ls-02 (previous step)

i < 0, band(j, 31) < 32, 0 < band(j, 31), 0 < (~1 * 2 ^ band(j, 31) + i + 1) / 2 ^ band(j, 31) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ i % j = 0 ∧ i < 0 ∧ i0 < 0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j
Proof step 11:
(~1 * 2 ^ band(j, 31)) % 2 ^ band(j, 31) = 0 ∧ ~1 * 2 ^ band(j, 31) < 0 ∧ i + 1 < 0 ∧ 0 < 2 ^ band(j, 31), i < 0, band(j, 31) < 32, 0 < band(j, 31), 0 < (~1 * 2 ^ band(j, 31)) / 2 ^ band(j, 31) + (i + 1) / 2 ^ band(j, 31) ⊦

Rule argument

seq: ⊦ i % j = 0 ∧ i < 0 ∧ i0 < 0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j subst: [j, i0, i] → [2 ^ band(j, 31), i + 1, ~1 * 2 ^ band(j, 31)] rotate: F spec: int-div inst: name: div-add1-negneg

Additional information

Used simplifier rules: band-commutativity: ⊦ band(i, j) = band(j, i);%band-ls-zero-01: ⊦ ¬ band(j, 31) < 0;%band-associativity: ⊦ band(i, band(i0, j)) = band(band(i, i0), j);% ⊦ i * ~1 = ~ i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ i + ~ j = i - j;% ⊦ i + (j - i0) = i + j - i0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ 0 < j → (i ^ j = 0 ↔ i = 0);%j ≠ 0 ⊦ ~ i % j = ~(i % j);%i ≠ 0 ⊦ i % i = 0;% ⊦ ~ i < 0 ↔ 0 < i;% ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i + (* j) < (* j0) ↔ i < j0 - j;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < 0 → (~1 ≤ i ↔ i = ~1);% ⊦ 0 - i = ~ i;%j ≠ 0 ⊦ ~ i / j = ~(i / j);%i ≠ 0 ⊦ i / i = 1; Used terms: 2 ^ band(j, 31), i + 1, ~1 * 2 ^ band(j, 31)